# arith2geom

Arithmetic to geometric moments of asset returns

## Description

example

[mg,Cg = arith2geom(ma,Ca) transforms moments associated with a simple Brownian motion into equivalent continuously compounded moments associated with a geometric Brownian motion with a possible change in periodicity.

example

[mg,Cg = arith2geom(___,t) adds an optional argument t.

## Examples

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This example shows several variations of using arith2geom.

Given arithmetic mean m and covariance C of monthly total returns, obtain annual geometric mean mg and covariance Cg. In this case, the output period (1 year) is 12 times the input period (1 month) so that the optional input t = 12.

m = [ 0.05; 0.1; 0.12; 0.18 ];
C = [ 0.0064 0.00408 0.00192 0;
0.00408 0.0289 0.0204 0.0119;
0.00192 0.0204 0.0576 0.0336;
0 0.0119 0.0336 0.1225 ];
[mg, Cg] = arith2geom(m, C, 12)
mg = 4×1

0.8934
2.9488
4.9632
17.0835

Cg = 4×4
103 ×

0.0003    0.0004    0.0003         0
0.0004    0.0065    0.0065    0.0110
0.0003    0.0065    0.0354    0.0536
0    0.0110    0.0536    1.0952

Given annual arithmetic mean m and covariance C of asset returns, obtain monthly geometric mean mg and covariance Cg. In this case, the output period (1 month) is 1/12 times the input period (1 year) so that the optional input t = 1/12.

[mg, Cg] = arith2geom(m, C, 1/12)
mg = 4×1

0.0044
0.0096
0.0125
0.0203

Cg = 4×4

0.0005    0.0003    0.0002         0
0.0003    0.0025    0.0017    0.0010
0.0002    0.0017    0.0049    0.0029
0    0.0010    0.0029    0.0107

Given arithmetic mean m and covariance C of monthly total returns, obtain quarterly continuously compounded return moments. In this case, the output is 3 of the input periods so that the optional input t = 3.

[mg, Cg] = arith2geom(m, C, 3)
mg = 4×1

0.1730
0.4097
0.5627
1.0622

Cg = 4×4

0.0267    0.0204    0.0106         0
0.0204    0.1800    0.1390    0.1057
0.0106    0.1390    0.4606    0.3418
0    0.1057    0.3418    1.8886

## Input Arguments

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Arithmetic mean of asset-return data, specified as an n-vector.

Data Types: double

Arithmetic covariance of asset-return data, specified as an n-by-n symmetric, positive semidefinite matrix. If Ca is not a symmetric positive semidefinite matrix, use nearcorr to create a positive semidefinite matrix for a correlation matrix.

Data Types: double

(Optional) Target period of geometric moments in terms of periodicity of arithmetic moments, specified as a scalar positive numeric.

Data Types: double

## Output Arguments

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Continuously compounded or "geometric" mean of asset returns over the target period (t), returned as an n-vector.

Continuously compounded or "geometric" covariance of asset returns over the target period (t), returned as an n-by-n matrix.

## Algorithms

Arithmetic returns over period tA are modeled as multivariate normal random variables with moments

$E\left[\text{X}\right]={\text{m}}_{A}$

and

$\mathrm{cov}\left(\text{X}\right)={\text{C}}_{A}$

Geometric returns over period tG are modeled as multivariate lognormal random variables with moments

$E\left[\text{Y}\right]=1+{\text{m}}_{G}$

$\mathrm{cov}\left(\text{Y}\right)={\text{C}}_{G}$

Given t = tG / tA, the transformation from geometric to arithmetic moments is

$1+{\text{m}}_{{G}_{i}}=\mathrm{exp}\left(t{\text{m}}_{{A}_{i}}+\frac{1}{2}t{\text{C}}_{{A}_{ii}}\right)$

${\text{C}}_{{G}_{ij}}=\left(1+{\text{m}}_{{G}_{i}}\right)\left(1+{\text{m}}_{{G}_{\text{j}}}\right)\left(\mathrm{exp}\left(t{\text{C}}_{A}{}_{ij}\right)-1\right)$

For i,j = 1,..., n.

Note

If t = 1, then Y = exp(X).

The arith2geom function has no restriction on the input mean ma but requires the input covariance Ca to be a symmetric positive-semidefinite matrix.

The functions arith2geom and geom2arith are complementary so that, given m, C, and t, the sequence

[mg,Cg] = arith2geom(m,C,t);
[ma,Ca] = geom2arith(mg,Cg,1/t);

yields ma = m and Ca = C.

## Version History

Introduced before R2006a